TEST CODE 01234020FORM TP 2007017 JANUARY 2007

CARIBBEAN EXAMINATIONS COUNCILSECONDARY EDUCATION CERTIFICATE

EXAMINA TIONMATHEMATICS

Paper 02 - General Proficiency

2 hours 40 minutes

( 03 JANUARY 2007 (a.m.))

INSTRUCTIONS TO CANDIDATES

1. Answer ALL questions in Section I, and ANY TWO in Section H.

2. Write your answers in the booklet provided.

3. All working must be shown clearly.

4. A list of formulae is provided on page 2 of this booklet.

Examination Materials

Electronic calculator (non-programmable)Geometry setMathematical tables (provided)Graph paper (provided)

DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO

Copyright © 2005 Caribbean Examinations Council®.All rights reserved.

012340201JANUARYIF 2007

Page 2

LIST OF FORMULAE

Volume of a prism V = AIz where A is the area of a cross-section and h is the perpendicularlength.

Volume of cy linder V= n?h where r is the radius of the base and h is the perpendicular height.

Volume of a right pyramid V = } Ah where A is the area of the base and h is the perpendicular height.

Circumference C = 2nr where r is the radius of the circle.

Area of a circle A = n?where r is the radius of the circle.

Area of trapezium A = ~ (a + b) h where a and b are the lengths of the parallel sides and his

the perpendicular distance between the parallel sides.

Roots of quadratic equations If ax' + bx + c = 0,

then x = -b ± ~b2 -4ac2a

Trigonometric ratios sin o opposite side= hypotenuse

cos e adjacent side= hypotenuse

tan e opposite side= adjacent side

Opposite

Adjacent

Area of triangle Area of 11 = ~ bh where b is the length of the base and h is the

perpendicular height~

Area of MBC = 1ab sin C ~( b )

Area of MBC = ~s (s - a) (s - b) (s - c)

where s = a + b + c2

B

Sine rule asin A = b

sin Bc= sin C

Cosine rule a2 = b2 + c2 - 2bc cos ACL---------~--------~A

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Page 3

SECTION I

Answer ALL the questions in this section.

All working must be clearly shown.

1. (a) Using a calculator, or otherwise, evaluate

(i) 5.24 (4 - 1.67) ( 2 marks)

(ii)1.68

( 3 marks)21.5 - 1.45

(b) A sum of money is shared between Aaron and Betty in the ratio 2: 5. Aaron received$60. How much money was shared altogether? ( 3 marks)

(c) In St. Vincent, 3 litres of gasoline cost EC$lOAO.

(i) Calculate the cost of 5 litres of gasoline in St. Vincent, stating your answercorrect to the nearest cent. ( 2 marks)

(ii) How many litres of gasoline can be bought for EC $50.00 in St. Vincent?

Give your answer correct to the nearest whole number. ( 2 marks)

Total 12 marks

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Page 4

2. (a) If a = 2, b = -3 and e = 4, evaluate

(i) ab - be ( 1 mark)

(ii) b(a - e)2 ( 2 marks)

(b) Solve for x where x E Z:

(i)x x-+-=52 3

( 3 marks)

(ii) 4-x~13 3 marks)

(c) The cost of ONE muffin is $m.The cost of THREE cupcakes is $2m.

(i) Write an algebraic expression in m for the cost of:

a) FIVE muffins 1mark)

b) SIX cupcakes ( 1mark)

(ii) Write an equation, in terms of m, to represent the following information.

The TOTAL cost of 5 muffins and 6 cupcakes is $31.50. ( 1mark)

Total 12 marks

3. (a) Describe, using set notation only, the shaded regions in each Venn diagram below. Thefirst one is done for you.

(i) U~-------------. r--------------,u

AnB( 3 marks)

(b) The following information is given.

u= {l,2,3,4,5,6,7,8,9,1O}

p = {prime numbers}

Q = {odd numbers}

Draw a Venn diagram to represent the information above. ( 3 marks)

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Page 5

(c) The Venn diagram below shows the number of elements in each region.

u

8

Determine how many elements are in EACH of the following sets:

(i) AuB ( 1 mark)

(ii) AnB ( 1 mark)

(iii) (A nBY ( 1 mark)

(iv) u ( 1 mark)

Total 10 marks

4. (a) (i) Using a pencil, ruler and a pair of compasses only, construct ~ ABCwithBC= 6 cm and AB =AC = 8 cm. ( 3 marks)

An construction lines must be dearly shown.

(ii) Draw a line segment AD such that AD meets BC at D and is perpendicular toBC. ( 2 marks)

(iii) Measure and state

a) the length of the line segment AD ( 1 mark)

b) the size of angle ABC ( 1mark)

(b) P is the point (2, 4) and Q is the point (6, 10).

Calculate

(i) the gradient of PQ ( 2 marks)

(ii) the midpoint of PQ. ( 2 marks)

Total 11 marks

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Page 6

5. An answer sheet is provided for this question.

(a) 1and g are functions defined as follows

fx~7x+4

1g:x~ 2x

Calculate

(i) g (3) ( 1 mark)

(ii) 1(-2)

(iii) 1-\11)

( 2 marks)

( 2 marks)

(b) On the answer sheet provided, ~ ABC is mapped onto ~ A 'B'C' under a reflection.

(i) Write down the equation of the mirror line. ( 1 mark)

~A 'B'C'is mapped onto ~A''B''C''by a rotation of 1800 about the point (5, 4).

(ii) Determine the coordinates of the vertices of ~ A ''B''C''. ( 3 marks)

(iii) State the transformation that maps ~ ABC onto ~ A ''B''C''. ( 2 marks)

Total 11 marks

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Page 7

6. The table below shows a frequency distribution of the scores of 100 students in an examination.

Scores Frequency CumullativeFrequency

21 - 25 5 5

26 - 30 18

31 - 35 23

36 - 40 22

41 - 45 21

46 - 50 11 100

(i) Copy and complete the table above to show the cumulative frequency for thedistribution. ( 2 marks)

(ii) Using a scale of 2 cm to represent a score of 5 on the horizontal axis and a scale of2 cm to represent 10 students on the vertical axis, draw a cumulative frequency curveof the scores. Start your horizontal scale at 20. ( 6 marks)

(iii) Using the cumulative frequency curve, determine the median score for the distribution.( 2 marks)

(iv) What is the probability that a student chosen at random has a score greater than 40?( 2 marks)

Total 12 marks

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Page 8

7. (a) The diagram below, not drawn to scale, shows a prism of length 30 cm. Thecross-section WXYZ is a square with area 144 cm'.

Calculate

(i) the volume, in crrr', of the prism ( 2 marks)

(ii) the total surface area, in cm2, of the prism. 4 marks)

(b) The diagram below, not drawn to scale, shows the sector of a circle with centre O.£MON = 45° and ON = 15 cm.

M N

Use 'Tt = 3.14

o

Calculate, giving your answer correct to 2 decimal places

(i) the length of the minor arc MN ( 2 marks)

(ii) the perimeter of the figure MON ( 2 marks)

(iii) the area of the figure MON. 2 marks)

Tota) 12 marks

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Page 9

8. A large equilateral triangle is subdivided into a set of smaller equilateral triangles by thefollowing procedure:

The midpoints of the sides of each equilateral triangle are joined to form a new set of smaller triangles.

The procedure is repeated my times.

The table below shows the results when the above procedure has been repeated twice, that is,when n = 2.

n Result after each step No. of trianglesformed

0 A 1

/\1 AA 4

AA2 /\/\/\ 16

/\/\/\/\3 (i)

6 (ii)

(iii) 65536

m (iv)

(i) Calculate the number of triangles formed when n = 3. ( 2 marks)

(ii) Determine the number of triangles formed when n = 6. ( 2 marks)

A shape has 65 536 small triangles.

(iii) Calculate the value of n. ( 3 marks)

(iv) Determine the number of small triangles in a shape after carrying out the procedurem times. ( 3 marks)

Total 10 marks

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9. (a)

SECTION II

Answer TWO questions in this section

ALGEBRA AND RELATIONS, FUNCTIONS AND GRAPHS

Factorise completely

(i)

(ii)

2p2 - 7p + 3

5p + 5q + p2 - l

Page 10

( 1 mark)

( 2 marks)

(b) Expand (x + 3)2 (x - 4), writing your answer in descending powers of x.

Givenjtx) = 2x2 + 4x ~ 5

(i) write fix) in the form/ex) = a(x + b)2 + c where a, b, c E R

(c)

(ii)

(iii)

(iv)

(v)

state the equation of the axis of symmetry

state the coordinates of the minimum point

sketch the graph of/ex)

on the graph offix) show clearly

a)

b)

the minimum point

the axis of symmetry.

( 3 marks)

( 3 marks)

( 1 mark)

( 1 mark)

( 2 marks)

( 1 mark)

( 1 mark)

Total 15 marks

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Page 11

10. An answer sheet is provided for this question.

Pam visits the stationery store where she intends to buy x pens and y pencils.

(a) Pam must buy at least 3 pens.

(i) Write an inequality to represent this information. ( 1mark)

The TOTAL number of pens and pencils must NOT be more than 10.

(ii) Write an inequality to represent this information. ( 2 marks)

EACH pen costs $5.00 and EACH pencil costs $2.00. More information about thepens and pencils is represented by:

·5x + 2y :::;35

(iii) Write the information represented by this inequality as a sentence in your ownwords. ( 2 marks)

(b) (i) On the answer sheet provided, draw the graph of the TWO inequalitiesobtained in (a) (i) and (a) (ii) above. ( 3 marks)

(ii). Write the coordinates of the vertices of the region that satisfies the fourinequalities (including y 2 0). ( 2 marks)

(c) Pam sells the x pens and y pencils and makes a profit of $1.50 on EACH pen and$1.00 on EACH pencil.

(i) Write an expression in x and y to represent the profit Pam makes. ( 1mark)

(ii) Calculate the maximum profit Pam makes. ( 2 marks)

(iii) If Pam buys 4 pens, show on your graph the maximum number of pencils shecan buy. ( 2 marks)

Total 15 marks

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Page 12

GEOMETRY AND TRIGONOMETRY

11. (a) Two circles with centres P and Q and radii 5 cm and 2 cm respectively are drawn sothat they touch each other at T and a straight line XY at Sand R.

x--------------~~~~------~~C----------------ys R

(i) State, with a reason,

a) why PTQ is a straight line

the length PQ

why PS is parallel to QR.

( 2 marks)

( 2 marks)

( 2 marks)

b)

c)

(ii) N is a point on PS such that QN is perpendicular to PS.

Calculate

a)

b)

the length PN

the length RS.

( 2 marks)

( 2 marks)

(b) In the diagram below, not drawn to scale, 0 is the centre of the circle. The measure ofangle LOM is 110°.

N

Calculate, giving reasons for your answers, the size of EACH of the following angles

(i)

(ii)

L.MNL

LLMO

( 2 marks)

( 3 marks)

Total IS marks

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Page 13

12. A boat leaves a dock at point A and travels for a distance of 15 km to point B on a bearing of135°.

The boat then changes course and travels for a distance of 8 km to point C on a bearing of 060°.

(a) Illustrate the above information in a clearly labelled diagram. ( 2 marks)

The diagram should show the

(i) north direction ( 1 mark)

(ii) bearings 135° and 060° ( 2 marks)

(iii) distances 8 km and 15 km. ( 2 marks)

Cb) Calculate

(i) the distance AC ( 3 marks)

(ii) L.BCA ( 3 marks)

(iii) the bearing of A from C. ( 2 marks)

Total 15 marks

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Page 14

VECTORS AND MATRICES-)

13. In the diagram below, M is the midpoint of ON.

r

o M N

(a) (i) Sketch the diagram above in your answer booklet and insert the point X on oiJ-)1-)

such that OX = 3" OM. ( 1 mark)

(ii)-) -)

Produce PX to Q such that PX = 4 XQ. ( 1mark)

(b) Write the following in terms ofL and g,-)

(i) OM

(ii) Pi-)

(iii) QM

( 2 marks)

( 3 marks)

( 4 marks)

(c)-) -) -)

Show that PN = 2 PM + OP ( 4 marks)

Total 15 marks

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Page 15

14. (a) Given that 0 = [~ 49P

] is a singular matrix, determine the valuers) ofp.( 4 marks)

(b) Given the linear equations

2x+ 5y= 63x + 4y = 8

(i) Write the equations in the form AX = B where A, X and B are matrices.( 2 marks)

(ii) a) Calculate the determinant of the matrix A. ( 2 marks)

b)[

-4

Show thatA-I~ ~ ( 2 marks)

c) Use the matrix A-I to solve for x and y. ( 5 marks)

Total 15 marks

END OF TEST

012340201JANUARY/F 2007

TEST CODE 01234020FORM TP 2007017 JANUARY 2007

CARIBBEAN EXAMINATIONS COUNCIL

SECONDARY EDUCATION CERTIFICATEEXAMINA TION

MA THEMA TICS

Paper 02 - General Proficiency

Answer Sheet for Question 10 Candidate Number .

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ATTACH THIS ANSWER SHEET TO YOUR ANSWER BOOKLET

01 234020lJANAUARYIF 2007

TEST CODE 01234020FORM TP 2007017

CARIBBEAN EXAMINATIONSJANU ARY 2007

COUNCIL

SECONDARY EDUCATION CERTIFICATEEXAMINA TION

MA THEMA TICS

Paper 02 General Proficiency

Answer Sheet for Question 5 (b) Candidate Number .

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ATTACH THIS ANSWER SHEET TO YOUR ANSWER BOOKLET

01234020/JANAUARYIF 2007

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